Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

TL;DR

This paper develops unconditionally stable, second-order accurate numerical schemes for simulating solid state phase transformations driven by mechano-chemical spinodal decomposition, enabling large-scale 3D microstructure computations.

Contribution

It introduces novel unconditionally stable, second-order accurate time-integration schemes for coupled nonlinear PDEs in phase transformation models, including reduced formulations for faster computations.

Findings

Schemes demonstrate second-order accuracy and unconditional stability.

Numerical results validate the effectiveness of the proposed methods.

Applicable to a broad class of polynomial free energy density problems.

Abstract

We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin's theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.